We would like to end our investigation of Green’s relations with a description of the D- and J -classes. Recall that for a semigroup S, xDy if and only if we can find some c ∈ S, such that xRc and yLc, and xJ y if and only if S1xS1 = S1yS1. Our first task
will be to characterize the D-classes as we did for the R-, L-, and H-classes.
Theorem 6.5.1. Assume (r, P, c) 6= (s, Q, d). Then (r, P, c)D(s, Q, d) if and only if there exist elements a, b ∈ S that obey the following:
(a) ac = c and bd = d; (b) aPr1 ⊆ P and bQ1
s ⊆ Q;
(c) aPr1 and bQ1s are isomorphic as labeled digraphs and there exists a label- preserving isomorphism θ : aP1
r → bQ1s such that subdigraph bQ1s contains
a cycle connecting cθ and d.
Proof: Suppose (r, P, c)D(s, Q, d) and assume (r, P, c) 6= (s, Q, d). We wish to show parts (a) and (b). There exists an element (t, A, x) ∈ M(S, Σ) such that
(r, P, c)R(t, A, x) and (t, A, x)L(s, Q, d). From Proposition 6.1.1, we know t = r and A = P , and thus we replace t by r and A by P for the rest of this proof. Since (r, P, x)L(s, Q, d), by Proposition 6.2.2(a)and (b) there exists an a, b ∈ S such that ax = x, bd = d, aPr1 ⊆ P , and bQ1
s ⊆ Q. Because xRc, there exists a y ∈ S such that
xy = c. Thus ac = axy = xy = c. Thus we have (a) and (b).
Finally we prove (c). Since left multiplication by a fixes x, it also fixes every vertex accessible from x. By Proposition 6.1.1, we know there is a cycle C in P containing x and c. Combining this with the previous fact, the cycle C is contained in aPr1. Proposition 6.2.2(c) states that aP1
r and bQ1s are isomorphic as labeled subdigraphs
and there exists a labeled digraph isomorphism θ : aPr1 → bQ1
s that maps x to d. The
function θ also maps the cycle C to a cycle in bQ1
s; the latter contains the vertices
xθ = d and cθ. Thus we have obtained (c).
Conversely, let (r, P, c), (s, Q, d) ∈ M and a, b ∈ S satisfy (a), (b), and (c). From (c), we know there is a cycle in bQ1s containing x and cθ. Thus we can find words v, w ∈ Σ∗ such that Q contains the paths cθ −→ x and xv −→ cθ. Since θ is andw isomorphism mapping aPr1 to bQ1s, the existence of a cycle cθ −→ cθ in bQvw 1
s implies
that there exists a cycle c −→ c in aPvw 1
r and hence in P . This cycle contains the
vertices c and c · v. Thus appealing to Proposition 6.1.1, we have (r, P, c)R(r, P, c · v). Moreover, using condition (a), we see that a(c·v) = (ac)·v = c·v. Combining this with the second half of (a), we satisfy Proposition 6.2.2 (a). Proposition 6.2.2 (b) fol- lows from (b). Finally, from Lemma 6.2.1(a), we have that (c · v)θ = (cθ) · v = x. Thus we obtain Proposition 6.2.2(c) as well, whereupon we have that (r, P, c · v)L(s, Q, d). We conclude that (r, P, c)D(s, Q, d). We now wish to show that D = J for semigroup graph expansions. Our approach will be constructive and rely on the structure of graph expansion elements.
Theorem 6.5.2. Let (S, Σ, f ) be a semigroup system. Then in M(S; Σ), D = J . Proof: It is a basic result of semigroup theory that D ⊆ J . We wish to show the reverse containment. Suppose for some (r, P, c) 6= (s, Q, d) that (r, P, c)J (s, Q, d).
Then there exist elements (r, P, x), (s, Q, y), (t, C, w), and (u, D, z) such that:
(r, P, c) = (r, P, x)(s, Q, d)(t, C, w) (6.5.1) (s, Q, d) = (s, Q, y)(r, P, c)(u, D, z). (6.5.2) By continually inserting these equations into each other, we obtain the following for any i ∈ N:
(r, P, c) = (r, P, x)(s, Q, y)i
(r, P, c) (u, D, z)(t, C, w)i
(6.5.3) (s, Q, d) = (s, Q, y) (r, P, x)(s, Q, y)i(r, P, c) (u, D, z)(t, C, w)i(u, D, z). (6.5.4) Inspection of Equation 6.5.3 indicates that for all i ∈ N, we have (xy)i ∈ V (P ). (Alternatively, we could justify this observation using the same arguments as used in the proof for L-classes.) Since P is a finite digraph, xy is periodic. Let k and m be the smallest natural numbers such that (xy)k= (xy)k+m. From Lemma 5.1.1,
(r, P, x)(s, Q, y)k+m = (r, P, x)(s, Q, y)k+2m. (6.5.5) We wish to show that (r, P, c)(u, D, z)R(r, P, c). First, using Equations 6.5.3 and 6.5.5 we have that:
(r, P, c) = (r, P, x)(s, Q, y)k+2m(r, P, c) (u, D, z)(t, C, w)k+2m = (r, P, x)(s, Q, y)k+m(r, P, c) (u, D, z)(t, C, w)k+2m = (r, P, c) (u, D, z)(t, C, w)m
= (r, P, c)(u, d, z) (t, C, w) (u, D, z)(t, C, w)m−1.
and obviously we can obtain (r, P, c)(u, D, z) from (r, P, c) by multiplying the latter on the right by (u, D, z). Thus (r, P, c)(u, D, z)R(r, P, c).
Now we wish to show that (r, P, c)(u, D, z)L(s, Q, d). In this case, we use Equa- tions 6.5.2, 6.5.3, 6.5.4 and 6.5.5:
(r, P, c)(u, D, z) = (r, P, x)(s, Q, y)k+m(r, P, c) (u, D, z)(t, C, w)k+m(u, D, z) = (r, P, x)(s, Q, y)k+2m (r, P, c) (u, D, z)(t, C, w)k+m (u, D, z) = (r, P, x)(s, Q, y)m−1(r, P, x)(s, Q, y) (r, P, x)(s, Q, y)k+m(r, P, c) (u, D, z)(t, C, w)k+m(u, D, z) = (r, P, x)(s, Q, y)m−1(r, P, x)(s, Q, d).
Combining this result with Equation 6.5.2 shows that (r, P, c)(u, D, z)L(s, Q, d). We conclude that (r, P, c)D(s, Q, d). We now consider certain finiteness properties of J -classes. Like the L-classes, J -classes can be finite or infinite. We will soon give an example of a semigroup graph expansion that contains both types. Before doing that, we investigate the finite-J - above property. Given two elements x, y ∈ S, we say that x is J -above y (x ≥J y) if
S1yS1 ⊆ S1xS1. This is equivalent to the existence of some a, b ∈ S such that y = axb.
A semigroup is called finite-J -above, if for each y ∈ S, the set {x ∈ S |x ≥J y} is
finite. In the following proposition we show that if a semigroup is finite-J -above, then its graph expansion (for any system) is as well. We note that Elston proves the same result for the semigroup Cayley expansion, but uses properties of derived categories to obtain it [3].
Proposition 6.5.3. Let (S, Σ, f ) be a semigroup system. If S is finite-J -above and Σ is finite, then M(S; Σ) is finite-J -above.
Proof: Suppose S is finite-J -above. Let (r, P, c) ∈ M(S; Σ). Consider the set X = {(s, Q, d)|(s, Q, d) ≥J (r, P, c)}.
If X is finite, we are done. By way of contradiction, suppose X is infinite. We make two observations:
1. There are a finite number of elements of S that are chosen vertices for elements of X. This is because all these elements are J -above c and we assumed that S is finite-J -above.
2. All roots of elements of X come from a finite subset of Σ. This is because P is finite and if (s, Q, d) ∈ X with (s, Q, d) 6= (r, P, c), then P contains an s-labeled edge.
Combining these two observations with our assumption that X is infinite, we know there exists some s ∈ Σ and d ∈ S for which there are an infinite number of graphs Qi such that (s, Qi, d) ∈ X, i ∈ N. We note that for each of these elements, there
exists some (r, Ai, ai), (ti, Bi, bi) such that
(r, P, c) = (r, Ai, ai)(s, Qi, d)(ti, Bi, bi).
Note that for each i, we know that ai ∈ V (P ). Since P is finite, there is some
a ∈ V (P ) for which there are an infinite number of ai with ai = a. We thus specify a
new subset:
Xa= {(s, Qi, d)|a = ai}.
Note that if (s, Qi, d) ∈ Xa, then a(Qi)1s ⊆ P . Construct the graph
Γ = [
(s,Qi,d)∈Xa
Qi.
We claim that V (Γ) is infinite. To see this, observe that Γ is the union of an infinite number of distinct graphs. However, the edge label set of Γ is finite because it is contained in the edge label set of P . Moreover, since Γ is a subset of the Cayley digraph, each vertex has at most one edge of each label emerging from it. Thus V (Γ) must be infinite. On the other hand, aΓ1s ⊆ P . For each vertex v ∈ V (P ), we form
the set Yv = {y ∈ V (Γ)|ay = v}. Since P is finite, there exists some v for which Yv is
infinite. However every y ∈ Yv is J -above v since ay1 = v. This contradicts the fact
that every element of S is finite-J -above. Thus, the assumption that X is infinite is incorrect. We conclude that M(S; Σ) is finite-J -above. Not all graph expansions are finite-J -above. Example 6.6 in the next section has two infinite J -classes.